Properties

Label 5520.2.a.bk.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.24264 q^{11} -5.41421 q^{13} -1.00000 q^{15} -4.41421 q^{17} +7.07107 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.24264 q^{29} +10.6569 q^{31} +2.24264 q^{33} +1.00000 q^{35} -3.00000 q^{37} +5.41421 q^{39} -1.58579 q^{41} -2.00000 q^{43} +1.00000 q^{45} -5.41421 q^{47} -6.00000 q^{49} +4.41421 q^{51} -6.41421 q^{53} -2.24264 q^{55} -7.07107 q^{57} +12.0711 q^{59} +1.07107 q^{61} +1.00000 q^{63} -5.41421 q^{65} -10.6569 q^{67} -1.00000 q^{69} -2.07107 q^{71} +15.5563 q^{73} -1.00000 q^{75} -2.24264 q^{77} -5.65685 q^{79} +1.00000 q^{81} -16.0711 q^{83} -4.41421 q^{85} -1.24264 q^{87} -10.8284 q^{89} -5.41421 q^{91} -10.6569 q^{93} +7.07107 q^{95} -1.17157 q^{97} -2.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} - 8 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 10 q^{31} - 4 q^{33} + 2 q^{35} - 6 q^{37} + 8 q^{39} - 6 q^{41} - 4 q^{43} + 2 q^{45} - 8 q^{47} - 12 q^{49} + 6 q^{51} - 10 q^{53} + 4 q^{55} + 10 q^{59} - 12 q^{61} + 2 q^{63} - 8 q^{65} - 10 q^{67} - 2 q^{69} + 10 q^{71} - 2 q^{75} + 4 q^{77} + 2 q^{81} - 18 q^{83} - 6 q^{85} + 6 q^{87} - 16 q^{89} - 8 q^{91} - 10 q^{93} - 8 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.24264 −0.676182 −0.338091 0.941113i \(-0.609781\pi\)
−0.338091 + 0.941113i \(0.609781\pi\)
\(12\) 0 0
\(13\) −5.41421 −1.50163 −0.750816 0.660511i \(-0.770340\pi\)
−0.750816 + 0.660511i \(0.770340\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −4.41421 −1.07060 −0.535302 0.844661i \(-0.679802\pi\)
−0.535302 + 0.844661i \(0.679802\pi\)
\(18\) 0 0
\(19\) 7.07107 1.62221 0.811107 0.584898i \(-0.198865\pi\)
0.811107 + 0.584898i \(0.198865\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.24264 0.230753 0.115376 0.993322i \(-0.463193\pi\)
0.115376 + 0.993322i \(0.463193\pi\)
\(30\) 0 0
\(31\) 10.6569 1.91403 0.957014 0.290043i \(-0.0936695\pi\)
0.957014 + 0.290043i \(0.0936695\pi\)
\(32\) 0 0
\(33\) 2.24264 0.390394
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 0 0
\(39\) 5.41421 0.866968
\(40\) 0 0
\(41\) −1.58579 −0.247658 −0.123829 0.992304i \(-0.539517\pi\)
−0.123829 + 0.992304i \(0.539517\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −5.41421 −0.789744 −0.394872 0.918736i \(-0.629211\pi\)
−0.394872 + 0.918736i \(0.629211\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 4.41421 0.618114
\(52\) 0 0
\(53\) −6.41421 −0.881060 −0.440530 0.897738i \(-0.645209\pi\)
−0.440530 + 0.897738i \(0.645209\pi\)
\(54\) 0 0
\(55\) −2.24264 −0.302398
\(56\) 0 0
\(57\) −7.07107 −0.936586
\(58\) 0 0
\(59\) 12.0711 1.57152 0.785760 0.618532i \(-0.212272\pi\)
0.785760 + 0.618532i \(0.212272\pi\)
\(60\) 0 0
\(61\) 1.07107 0.137136 0.0685681 0.997646i \(-0.478157\pi\)
0.0685681 + 0.997646i \(0.478157\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −5.41421 −0.671551
\(66\) 0 0
\(67\) −10.6569 −1.30194 −0.650971 0.759103i \(-0.725638\pi\)
−0.650971 + 0.759103i \(0.725638\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −2.07107 −0.245791 −0.122895 0.992420i \(-0.539218\pi\)
−0.122895 + 0.992420i \(0.539218\pi\)
\(72\) 0 0
\(73\) 15.5563 1.82073 0.910366 0.413803i \(-0.135800\pi\)
0.910366 + 0.413803i \(0.135800\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −2.24264 −0.255573
\(78\) 0 0
\(79\) −5.65685 −0.636446 −0.318223 0.948016i \(-0.603086\pi\)
−0.318223 + 0.948016i \(0.603086\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0711 −1.76403 −0.882014 0.471222i \(-0.843813\pi\)
−0.882014 + 0.471222i \(0.843813\pi\)
\(84\) 0 0
\(85\) −4.41421 −0.478789
\(86\) 0 0
\(87\) −1.24264 −0.133225
\(88\) 0 0
\(89\) −10.8284 −1.14781 −0.573905 0.818922i \(-0.694572\pi\)
−0.573905 + 0.818922i \(0.694572\pi\)
\(90\) 0 0
\(91\) −5.41421 −0.567564
\(92\) 0 0
\(93\) −10.6569 −1.10506
\(94\) 0 0
\(95\) 7.07107 0.725476
\(96\) 0 0
\(97\) −1.17157 −0.118955 −0.0594776 0.998230i \(-0.518943\pi\)
−0.0594776 + 0.998230i \(0.518943\pi\)
\(98\) 0 0
\(99\) −2.24264 −0.225394
\(100\) 0 0
\(101\) 2.75736 0.274368 0.137184 0.990546i \(-0.456195\pi\)
0.137184 + 0.990546i \(0.456195\pi\)
\(102\) 0 0
\(103\) 10.4853 1.03315 0.516573 0.856243i \(-0.327208\pi\)
0.516573 + 0.856243i \(0.327208\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −3.24264 −0.313478 −0.156739 0.987640i \(-0.550098\pi\)
−0.156739 + 0.987640i \(0.550098\pi\)
\(108\) 0 0
\(109\) −18.3848 −1.76094 −0.880471 0.474100i \(-0.842774\pi\)
−0.880471 + 0.474100i \(0.842774\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) −0.757359 −0.0712464 −0.0356232 0.999365i \(-0.511342\pi\)
−0.0356232 + 0.999365i \(0.511342\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −5.41421 −0.500544
\(118\) 0 0
\(119\) −4.41421 −0.404650
\(120\) 0 0
\(121\) −5.97056 −0.542778
\(122\) 0 0
\(123\) 1.58579 0.142986
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −11.4142 −1.01285 −0.506424 0.862285i \(-0.669033\pi\)
−0.506424 + 0.862285i \(0.669033\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −5.65685 −0.494242 −0.247121 0.968985i \(-0.579484\pi\)
−0.247121 + 0.968985i \(0.579484\pi\)
\(132\) 0 0
\(133\) 7.07107 0.613139
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −0.485281 −0.0414604 −0.0207302 0.999785i \(-0.506599\pi\)
−0.0207302 + 0.999785i \(0.506599\pi\)
\(138\) 0 0
\(139\) −2.51472 −0.213296 −0.106648 0.994297i \(-0.534012\pi\)
−0.106648 + 0.994297i \(0.534012\pi\)
\(140\) 0 0
\(141\) 5.41421 0.455959
\(142\) 0 0
\(143\) 12.1421 1.01538
\(144\) 0 0
\(145\) 1.24264 0.103196
\(146\) 0 0
\(147\) 6.00000 0.494872
\(148\) 0 0
\(149\) −10.5858 −0.867221 −0.433611 0.901100i \(-0.642761\pi\)
−0.433611 + 0.901100i \(0.642761\pi\)
\(150\) 0 0
\(151\) 0.343146 0.0279248 0.0139624 0.999903i \(-0.495555\pi\)
0.0139624 + 0.999903i \(0.495555\pi\)
\(152\) 0 0
\(153\) −4.41421 −0.356868
\(154\) 0 0
\(155\) 10.6569 0.855979
\(156\) 0 0
\(157\) 1.48528 0.118538 0.0592692 0.998242i \(-0.481123\pi\)
0.0592692 + 0.998242i \(0.481123\pi\)
\(158\) 0 0
\(159\) 6.41421 0.508680
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −7.65685 −0.599731 −0.299866 0.953981i \(-0.596942\pi\)
−0.299866 + 0.953981i \(0.596942\pi\)
\(164\) 0 0
\(165\) 2.24264 0.174589
\(166\) 0 0
\(167\) 11.0711 0.856705 0.428352 0.903612i \(-0.359094\pi\)
0.428352 + 0.903612i \(0.359094\pi\)
\(168\) 0 0
\(169\) 16.3137 1.25490
\(170\) 0 0
\(171\) 7.07107 0.540738
\(172\) 0 0
\(173\) −11.3137 −0.860165 −0.430083 0.902790i \(-0.641516\pi\)
−0.430083 + 0.902790i \(0.641516\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −12.0711 −0.907317
\(178\) 0 0
\(179\) 12.3431 0.922570 0.461285 0.887252i \(-0.347389\pi\)
0.461285 + 0.887252i \(0.347389\pi\)
\(180\) 0 0
\(181\) −6.14214 −0.456541 −0.228271 0.973598i \(-0.573307\pi\)
−0.228271 + 0.973598i \(0.573307\pi\)
\(182\) 0 0
\(183\) −1.07107 −0.0791756
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) 9.89949 0.723923
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 13.8995 1.00573 0.502866 0.864364i \(-0.332279\pi\)
0.502866 + 0.864364i \(0.332279\pi\)
\(192\) 0 0
\(193\) 7.51472 0.540921 0.270461 0.962731i \(-0.412824\pi\)
0.270461 + 0.962731i \(0.412824\pi\)
\(194\) 0 0
\(195\) 5.41421 0.387720
\(196\) 0 0
\(197\) 13.7990 0.983137 0.491569 0.870839i \(-0.336424\pi\)
0.491569 + 0.870839i \(0.336424\pi\)
\(198\) 0 0
\(199\) −23.4558 −1.66274 −0.831370 0.555719i \(-0.812443\pi\)
−0.831370 + 0.555719i \(0.812443\pi\)
\(200\) 0 0
\(201\) 10.6569 0.751677
\(202\) 0 0
\(203\) 1.24264 0.0872163
\(204\) 0 0
\(205\) −1.58579 −0.110756
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −15.8579 −1.09691
\(210\) 0 0
\(211\) −10.1716 −0.700240 −0.350120 0.936705i \(-0.613859\pi\)
−0.350120 + 0.936705i \(0.613859\pi\)
\(212\) 0 0
\(213\) 2.07107 0.141907
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 10.6569 0.723434
\(218\) 0 0
\(219\) −15.5563 −1.05120
\(220\) 0 0
\(221\) 23.8995 1.60765
\(222\) 0 0
\(223\) 26.9706 1.80608 0.903041 0.429554i \(-0.141329\pi\)
0.903041 + 0.429554i \(0.141329\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) 6.82843 0.451235 0.225618 0.974216i \(-0.427560\pi\)
0.225618 + 0.974216i \(0.427560\pi\)
\(230\) 0 0
\(231\) 2.24264 0.147555
\(232\) 0 0
\(233\) −18.3431 −1.20170 −0.600850 0.799362i \(-0.705171\pi\)
−0.600850 + 0.799362i \(0.705171\pi\)
\(234\) 0 0
\(235\) −5.41421 −0.353184
\(236\) 0 0
\(237\) 5.65685 0.367452
\(238\) 0 0
\(239\) 6.07107 0.392705 0.196352 0.980533i \(-0.437090\pi\)
0.196352 + 0.980533i \(0.437090\pi\)
\(240\) 0 0
\(241\) 3.75736 0.242033 0.121016 0.992651i \(-0.461385\pi\)
0.121016 + 0.992651i \(0.461385\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.00000 −0.383326
\(246\) 0 0
\(247\) −38.2843 −2.43597
\(248\) 0 0
\(249\) 16.0711 1.01846
\(250\) 0 0
\(251\) −4.34315 −0.274137 −0.137068 0.990562i \(-0.543768\pi\)
−0.137068 + 0.990562i \(0.543768\pi\)
\(252\) 0 0
\(253\) −2.24264 −0.140994
\(254\) 0 0
\(255\) 4.41421 0.276429
\(256\) 0 0
\(257\) −12.7279 −0.793946 −0.396973 0.917830i \(-0.629939\pi\)
−0.396973 + 0.917830i \(0.629939\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 1.24264 0.0769175
\(262\) 0 0
\(263\) −29.7279 −1.83310 −0.916551 0.399918i \(-0.869039\pi\)
−0.916551 + 0.399918i \(0.869039\pi\)
\(264\) 0 0
\(265\) −6.41421 −0.394022
\(266\) 0 0
\(267\) 10.8284 0.662689
\(268\) 0 0
\(269\) −18.4142 −1.12273 −0.561367 0.827567i \(-0.689724\pi\)
−0.561367 + 0.827567i \(0.689724\pi\)
\(270\) 0 0
\(271\) −23.9706 −1.45611 −0.728054 0.685520i \(-0.759575\pi\)
−0.728054 + 0.685520i \(0.759575\pi\)
\(272\) 0 0
\(273\) 5.41421 0.327683
\(274\) 0 0
\(275\) −2.24264 −0.135236
\(276\) 0 0
\(277\) −11.6569 −0.700392 −0.350196 0.936676i \(-0.613885\pi\)
−0.350196 + 0.936676i \(0.613885\pi\)
\(278\) 0 0
\(279\) 10.6569 0.638009
\(280\) 0 0
\(281\) 9.07107 0.541135 0.270567 0.962701i \(-0.412789\pi\)
0.270567 + 0.962701i \(0.412789\pi\)
\(282\) 0 0
\(283\) 24.4558 1.45375 0.726875 0.686770i \(-0.240972\pi\)
0.726875 + 0.686770i \(0.240972\pi\)
\(284\) 0 0
\(285\) −7.07107 −0.418854
\(286\) 0 0
\(287\) −1.58579 −0.0936060
\(288\) 0 0
\(289\) 2.48528 0.146193
\(290\) 0 0
\(291\) 1.17157 0.0686788
\(292\) 0 0
\(293\) −33.0416 −1.93031 −0.965156 0.261674i \(-0.915725\pi\)
−0.965156 + 0.261674i \(0.915725\pi\)
\(294\) 0 0
\(295\) 12.0711 0.702805
\(296\) 0 0
\(297\) 2.24264 0.130131
\(298\) 0 0
\(299\) −5.41421 −0.313112
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 0 0
\(303\) −2.75736 −0.158406
\(304\) 0 0
\(305\) 1.07107 0.0613292
\(306\) 0 0
\(307\) 22.3848 1.27757 0.638783 0.769387i \(-0.279438\pi\)
0.638783 + 0.769387i \(0.279438\pi\)
\(308\) 0 0
\(309\) −10.4853 −0.596487
\(310\) 0 0
\(311\) 20.9706 1.18913 0.594566 0.804047i \(-0.297324\pi\)
0.594566 + 0.804047i \(0.297324\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 3.55635 0.199744 0.0998722 0.995000i \(-0.468157\pi\)
0.0998722 + 0.995000i \(0.468157\pi\)
\(318\) 0 0
\(319\) −2.78680 −0.156031
\(320\) 0 0
\(321\) 3.24264 0.180987
\(322\) 0 0
\(323\) −31.2132 −1.73675
\(324\) 0 0
\(325\) −5.41421 −0.300327
\(326\) 0 0
\(327\) 18.3848 1.01668
\(328\) 0 0
\(329\) −5.41421 −0.298495
\(330\) 0 0
\(331\) −30.7990 −1.69287 −0.846433 0.532496i \(-0.821254\pi\)
−0.846433 + 0.532496i \(0.821254\pi\)
\(332\) 0 0
\(333\) −3.00000 −0.164399
\(334\) 0 0
\(335\) −10.6569 −0.582246
\(336\) 0 0
\(337\) 7.65685 0.417095 0.208548 0.978012i \(-0.433126\pi\)
0.208548 + 0.978012i \(0.433126\pi\)
\(338\) 0 0
\(339\) 0.757359 0.0411341
\(340\) 0 0
\(341\) −23.8995 −1.29423
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −13.1716 −0.707087 −0.353544 0.935418i \(-0.615023\pi\)
−0.353544 + 0.935418i \(0.615023\pi\)
\(348\) 0 0
\(349\) −11.8284 −0.633161 −0.316581 0.948566i \(-0.602535\pi\)
−0.316581 + 0.948566i \(0.602535\pi\)
\(350\) 0 0
\(351\) 5.41421 0.288989
\(352\) 0 0
\(353\) −15.7574 −0.838680 −0.419340 0.907829i \(-0.637738\pi\)
−0.419340 + 0.907829i \(0.637738\pi\)
\(354\) 0 0
\(355\) −2.07107 −0.109921
\(356\) 0 0
\(357\) 4.41421 0.233625
\(358\) 0 0
\(359\) 11.7574 0.620530 0.310265 0.950650i \(-0.399582\pi\)
0.310265 + 0.950650i \(0.399582\pi\)
\(360\) 0 0
\(361\) 31.0000 1.63158
\(362\) 0 0
\(363\) 5.97056 0.313373
\(364\) 0 0
\(365\) 15.5563 0.814257
\(366\) 0 0
\(367\) 19.1421 0.999211 0.499606 0.866253i \(-0.333478\pi\)
0.499606 + 0.866253i \(0.333478\pi\)
\(368\) 0 0
\(369\) −1.58579 −0.0825527
\(370\) 0 0
\(371\) −6.41421 −0.333009
\(372\) 0 0
\(373\) −24.9706 −1.29293 −0.646463 0.762945i \(-0.723753\pi\)
−0.646463 + 0.762945i \(0.723753\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −6.72792 −0.346506
\(378\) 0 0
\(379\) 17.6569 0.906972 0.453486 0.891263i \(-0.350180\pi\)
0.453486 + 0.891263i \(0.350180\pi\)
\(380\) 0 0
\(381\) 11.4142 0.584768
\(382\) 0 0
\(383\) −26.0711 −1.33217 −0.666085 0.745876i \(-0.732031\pi\)
−0.666085 + 0.745876i \(0.732031\pi\)
\(384\) 0 0
\(385\) −2.24264 −0.114296
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) −36.1421 −1.83248 −0.916240 0.400631i \(-0.868791\pi\)
−0.916240 + 0.400631i \(0.868791\pi\)
\(390\) 0 0
\(391\) −4.41421 −0.223236
\(392\) 0 0
\(393\) 5.65685 0.285351
\(394\) 0 0
\(395\) −5.65685 −0.284627
\(396\) 0 0
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 0 0
\(399\) −7.07107 −0.353996
\(400\) 0 0
\(401\) −14.4853 −0.723360 −0.361680 0.932302i \(-0.617797\pi\)
−0.361680 + 0.932302i \(0.617797\pi\)
\(402\) 0 0
\(403\) −57.6985 −2.87417
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 6.72792 0.333491
\(408\) 0 0
\(409\) −28.4558 −1.40705 −0.703525 0.710670i \(-0.748392\pi\)
−0.703525 + 0.710670i \(0.748392\pi\)
\(410\) 0 0
\(411\) 0.485281 0.0239372
\(412\) 0 0
\(413\) 12.0711 0.593978
\(414\) 0 0
\(415\) −16.0711 −0.788898
\(416\) 0 0
\(417\) 2.51472 0.123146
\(418\) 0 0
\(419\) −36.3848 −1.77751 −0.888756 0.458380i \(-0.848430\pi\)
−0.888756 + 0.458380i \(0.848430\pi\)
\(420\) 0 0
\(421\) 14.8701 0.724722 0.362361 0.932038i \(-0.381971\pi\)
0.362361 + 0.932038i \(0.381971\pi\)
\(422\) 0 0
\(423\) −5.41421 −0.263248
\(424\) 0 0
\(425\) −4.41421 −0.214121
\(426\) 0 0
\(427\) 1.07107 0.0518326
\(428\) 0 0
\(429\) −12.1421 −0.586228
\(430\) 0 0
\(431\) 5.17157 0.249106 0.124553 0.992213i \(-0.460250\pi\)
0.124553 + 0.992213i \(0.460250\pi\)
\(432\) 0 0
\(433\) 24.1716 1.16161 0.580806 0.814042i \(-0.302738\pi\)
0.580806 + 0.814042i \(0.302738\pi\)
\(434\) 0 0
\(435\) −1.24264 −0.0595801
\(436\) 0 0
\(437\) 7.07107 0.338255
\(438\) 0 0
\(439\) 2.20101 0.105048 0.0525242 0.998620i \(-0.483273\pi\)
0.0525242 + 0.998620i \(0.483273\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 15.0711 0.716048 0.358024 0.933712i \(-0.383451\pi\)
0.358024 + 0.933712i \(0.383451\pi\)
\(444\) 0 0
\(445\) −10.8284 −0.513317
\(446\) 0 0
\(447\) 10.5858 0.500691
\(448\) 0 0
\(449\) −31.7279 −1.49733 −0.748667 0.662947i \(-0.769306\pi\)
−0.748667 + 0.662947i \(0.769306\pi\)
\(450\) 0 0
\(451\) 3.55635 0.167462
\(452\) 0 0
\(453\) −0.343146 −0.0161224
\(454\) 0 0
\(455\) −5.41421 −0.253822
\(456\) 0 0
\(457\) 19.0000 0.888783 0.444391 0.895833i \(-0.353420\pi\)
0.444391 + 0.895833i \(0.353420\pi\)
\(458\) 0 0
\(459\) 4.41421 0.206038
\(460\) 0 0
\(461\) 18.4853 0.860945 0.430473 0.902604i \(-0.358347\pi\)
0.430473 + 0.902604i \(0.358347\pi\)
\(462\) 0 0
\(463\) −28.0416 −1.30321 −0.651603 0.758561i \(-0.725903\pi\)
−0.651603 + 0.758561i \(0.725903\pi\)
\(464\) 0 0
\(465\) −10.6569 −0.494200
\(466\) 0 0
\(467\) 10.4142 0.481912 0.240956 0.970536i \(-0.422539\pi\)
0.240956 + 0.970536i \(0.422539\pi\)
\(468\) 0 0
\(469\) −10.6569 −0.492088
\(470\) 0 0
\(471\) −1.48528 −0.0684382
\(472\) 0 0
\(473\) 4.48528 0.206233
\(474\) 0 0
\(475\) 7.07107 0.324443
\(476\) 0 0
\(477\) −6.41421 −0.293687
\(478\) 0 0
\(479\) 9.89949 0.452319 0.226160 0.974090i \(-0.427383\pi\)
0.226160 + 0.974090i \(0.427383\pi\)
\(480\) 0 0
\(481\) 16.2426 0.740601
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −1.17157 −0.0531984
\(486\) 0 0
\(487\) −5.07107 −0.229792 −0.114896 0.993378i \(-0.536653\pi\)
−0.114896 + 0.993378i \(0.536653\pi\)
\(488\) 0 0
\(489\) 7.65685 0.346255
\(490\) 0 0
\(491\) −24.8995 −1.12370 −0.561849 0.827240i \(-0.689910\pi\)
−0.561849 + 0.827240i \(0.689910\pi\)
\(492\) 0 0
\(493\) −5.48528 −0.247045
\(494\) 0 0
\(495\) −2.24264 −0.100799
\(496\) 0 0
\(497\) −2.07107 −0.0929001
\(498\) 0 0
\(499\) 9.82843 0.439981 0.219990 0.975502i \(-0.429397\pi\)
0.219990 + 0.975502i \(0.429397\pi\)
\(500\) 0 0
\(501\) −11.0711 −0.494619
\(502\) 0 0
\(503\) −24.6985 −1.10125 −0.550626 0.834752i \(-0.685611\pi\)
−0.550626 + 0.834752i \(0.685611\pi\)
\(504\) 0 0
\(505\) 2.75736 0.122701
\(506\) 0 0
\(507\) −16.3137 −0.724517
\(508\) 0 0
\(509\) 15.7990 0.700278 0.350139 0.936698i \(-0.386134\pi\)
0.350139 + 0.936698i \(0.386134\pi\)
\(510\) 0 0
\(511\) 15.5563 0.688172
\(512\) 0 0
\(513\) −7.07107 −0.312195
\(514\) 0 0
\(515\) 10.4853 0.462037
\(516\) 0 0
\(517\) 12.1421 0.534011
\(518\) 0 0
\(519\) 11.3137 0.496617
\(520\) 0 0
\(521\) −5.07107 −0.222168 −0.111084 0.993811i \(-0.535432\pi\)
−0.111084 + 0.993811i \(0.535432\pi\)
\(522\) 0 0
\(523\) −30.3431 −1.32681 −0.663407 0.748259i \(-0.730890\pi\)
−0.663407 + 0.748259i \(0.730890\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) −47.0416 −2.04917
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.0711 0.523840
\(532\) 0 0
\(533\) 8.58579 0.371892
\(534\) 0 0
\(535\) −3.24264 −0.140192
\(536\) 0 0
\(537\) −12.3431 −0.532646
\(538\) 0 0
\(539\) 13.4558 0.579584
\(540\) 0 0
\(541\) 44.4853 1.91257 0.956286 0.292434i \(-0.0944651\pi\)
0.956286 + 0.292434i \(0.0944651\pi\)
\(542\) 0 0
\(543\) 6.14214 0.263584
\(544\) 0 0
\(545\) −18.3848 −0.787517
\(546\) 0 0
\(547\) 8.68629 0.371399 0.185700 0.982607i \(-0.440545\pi\)
0.185700 + 0.982607i \(0.440545\pi\)
\(548\) 0 0
\(549\) 1.07107 0.0457121
\(550\) 0 0
\(551\) 8.78680 0.374330
\(552\) 0 0
\(553\) −5.65685 −0.240554
\(554\) 0 0
\(555\) 3.00000 0.127343
\(556\) 0 0
\(557\) 25.5858 1.08410 0.542052 0.840345i \(-0.317647\pi\)
0.542052 + 0.840345i \(0.317647\pi\)
\(558\) 0 0
\(559\) 10.8284 0.457994
\(560\) 0 0
\(561\) −9.89949 −0.417957
\(562\) 0 0
\(563\) 3.24264 0.136661 0.0683305 0.997663i \(-0.478233\pi\)
0.0683305 + 0.997663i \(0.478233\pi\)
\(564\) 0 0
\(565\) −0.757359 −0.0318623
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 4.14214 0.173647 0.0868237 0.996224i \(-0.472328\pi\)
0.0868237 + 0.996224i \(0.472328\pi\)
\(570\) 0 0
\(571\) −30.7279 −1.28592 −0.642962 0.765898i \(-0.722295\pi\)
−0.642962 + 0.765898i \(0.722295\pi\)
\(572\) 0 0
\(573\) −13.8995 −0.580660
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −21.9411 −0.913421 −0.456711 0.889615i \(-0.650972\pi\)
−0.456711 + 0.889615i \(0.650972\pi\)
\(578\) 0 0
\(579\) −7.51472 −0.312301
\(580\) 0 0
\(581\) −16.0711 −0.666740
\(582\) 0 0
\(583\) 14.3848 0.595757
\(584\) 0 0
\(585\) −5.41421 −0.223850
\(586\) 0 0
\(587\) −14.0000 −0.577842 −0.288921 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 0 0
\(589\) 75.3553 3.10496
\(590\) 0 0
\(591\) −13.7990 −0.567615
\(592\) 0 0
\(593\) −26.8701 −1.10342 −0.551711 0.834036i \(-0.686025\pi\)
−0.551711 + 0.834036i \(0.686025\pi\)
\(594\) 0 0
\(595\) −4.41421 −0.180965
\(596\) 0 0
\(597\) 23.4558 0.959984
\(598\) 0 0
\(599\) 8.48528 0.346699 0.173350 0.984860i \(-0.444541\pi\)
0.173350 + 0.984860i \(0.444541\pi\)
\(600\) 0 0
\(601\) 0.313708 0.0127964 0.00639822 0.999980i \(-0.497963\pi\)
0.00639822 + 0.999980i \(0.497963\pi\)
\(602\) 0 0
\(603\) −10.6569 −0.433981
\(604\) 0 0
\(605\) −5.97056 −0.242738
\(606\) 0 0
\(607\) 14.0416 0.569932 0.284966 0.958538i \(-0.408018\pi\)
0.284966 + 0.958538i \(0.408018\pi\)
\(608\) 0 0
\(609\) −1.24264 −0.0503543
\(610\) 0 0
\(611\) 29.3137 1.18591
\(612\) 0 0
\(613\) 48.6274 1.96404 0.982021 0.188769i \(-0.0604499\pi\)
0.982021 + 0.188769i \(0.0604499\pi\)
\(614\) 0 0
\(615\) 1.58579 0.0639451
\(616\) 0 0
\(617\) 28.8995 1.16345 0.581725 0.813386i \(-0.302378\pi\)
0.581725 + 0.813386i \(0.302378\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −10.8284 −0.433832
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 15.8579 0.633302
\(628\) 0 0
\(629\) 13.2426 0.528019
\(630\) 0 0
\(631\) 31.8995 1.26990 0.634949 0.772554i \(-0.281021\pi\)
0.634949 + 0.772554i \(0.281021\pi\)
\(632\) 0 0
\(633\) 10.1716 0.404284
\(634\) 0 0
\(635\) −11.4142 −0.452959
\(636\) 0 0
\(637\) 32.4853 1.28711
\(638\) 0 0
\(639\) −2.07107 −0.0819302
\(640\) 0 0
\(641\) −21.0711 −0.832257 −0.416129 0.909306i \(-0.636613\pi\)
−0.416129 + 0.909306i \(0.636613\pi\)
\(642\) 0 0
\(643\) −31.8284 −1.25519 −0.627595 0.778540i \(-0.715961\pi\)
−0.627595 + 0.778540i \(0.715961\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) 42.7279 1.67981 0.839904 0.542735i \(-0.182611\pi\)
0.839904 + 0.542735i \(0.182611\pi\)
\(648\) 0 0
\(649\) −27.0711 −1.06263
\(650\) 0 0
\(651\) −10.6569 −0.417675
\(652\) 0 0
\(653\) 49.8406 1.95041 0.975207 0.221294i \(-0.0710281\pi\)
0.975207 + 0.221294i \(0.0710281\pi\)
\(654\) 0 0
\(655\) −5.65685 −0.221032
\(656\) 0 0
\(657\) 15.5563 0.606911
\(658\) 0 0
\(659\) 35.6985 1.39062 0.695308 0.718712i \(-0.255268\pi\)
0.695308 + 0.718712i \(0.255268\pi\)
\(660\) 0 0
\(661\) 18.4853 0.718994 0.359497 0.933146i \(-0.382948\pi\)
0.359497 + 0.933146i \(0.382948\pi\)
\(662\) 0 0
\(663\) −23.8995 −0.928179
\(664\) 0 0
\(665\) 7.07107 0.274204
\(666\) 0 0
\(667\) 1.24264 0.0481152
\(668\) 0 0
\(669\) −26.9706 −1.04274
\(670\) 0 0
\(671\) −2.40202 −0.0927290
\(672\) 0 0
\(673\) 32.8701 1.26705 0.633524 0.773723i \(-0.281608\pi\)
0.633524 + 0.773723i \(0.281608\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −8.89949 −0.342035 −0.171018 0.985268i \(-0.554706\pi\)
−0.171018 + 0.985268i \(0.554706\pi\)
\(678\) 0 0
\(679\) −1.17157 −0.0449608
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) −44.7279 −1.71147 −0.855733 0.517417i \(-0.826893\pi\)
−0.855733 + 0.517417i \(0.826893\pi\)
\(684\) 0 0
\(685\) −0.485281 −0.0185416
\(686\) 0 0
\(687\) −6.82843 −0.260521
\(688\) 0 0
\(689\) 34.7279 1.32303
\(690\) 0 0
\(691\) 0.627417 0.0238681 0.0119340 0.999929i \(-0.496201\pi\)
0.0119340 + 0.999929i \(0.496201\pi\)
\(692\) 0 0
\(693\) −2.24264 −0.0851909
\(694\) 0 0
\(695\) −2.51472 −0.0953887
\(696\) 0 0
\(697\) 7.00000 0.265144
\(698\) 0 0
\(699\) 18.3431 0.693801
\(700\) 0 0
\(701\) −29.6985 −1.12170 −0.560848 0.827919i \(-0.689525\pi\)
−0.560848 + 0.827919i \(0.689525\pi\)
\(702\) 0 0
\(703\) −21.2132 −0.800071
\(704\) 0 0
\(705\) 5.41421 0.203911
\(706\) 0 0
\(707\) 2.75736 0.103701
\(708\) 0 0
\(709\) 37.0711 1.39223 0.696117 0.717929i \(-0.254910\pi\)
0.696117 + 0.717929i \(0.254910\pi\)
\(710\) 0 0
\(711\) −5.65685 −0.212149
\(712\) 0 0
\(713\) 10.6569 0.399102
\(714\) 0 0
\(715\) 12.1421 0.454090
\(716\) 0 0
\(717\) −6.07107 −0.226728
\(718\) 0 0
\(719\) −0.414214 −0.0154476 −0.00772378 0.999970i \(-0.502459\pi\)
−0.00772378 + 0.999970i \(0.502459\pi\)
\(720\) 0 0
\(721\) 10.4853 0.390492
\(722\) 0 0
\(723\) −3.75736 −0.139738
\(724\) 0 0
\(725\) 1.24264 0.0461505
\(726\) 0 0
\(727\) 9.14214 0.339063 0.169532 0.985525i \(-0.445775\pi\)
0.169532 + 0.985525i \(0.445775\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.82843 0.326531
\(732\) 0 0
\(733\) −24.1716 −0.892797 −0.446399 0.894834i \(-0.647294\pi\)
−0.446399 + 0.894834i \(0.647294\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 23.8995 0.880349
\(738\) 0 0
\(739\) 3.14214 0.115585 0.0577927 0.998329i \(-0.481594\pi\)
0.0577927 + 0.998329i \(0.481594\pi\)
\(740\) 0 0
\(741\) 38.2843 1.40641
\(742\) 0 0
\(743\) 26.6274 0.976865 0.488433 0.872602i \(-0.337569\pi\)
0.488433 + 0.872602i \(0.337569\pi\)
\(744\) 0 0
\(745\) −10.5858 −0.387833
\(746\) 0 0
\(747\) −16.0711 −0.588010
\(748\) 0 0
\(749\) −3.24264 −0.118484
\(750\) 0 0
\(751\) 36.7279 1.34022 0.670110 0.742261i \(-0.266247\pi\)
0.670110 + 0.742261i \(0.266247\pi\)
\(752\) 0 0
\(753\) 4.34315 0.158273
\(754\) 0 0
\(755\) 0.343146 0.0124884
\(756\) 0 0
\(757\) −47.0000 −1.70824 −0.854122 0.520073i \(-0.825905\pi\)
−0.854122 + 0.520073i \(0.825905\pi\)
\(758\) 0 0
\(759\) 2.24264 0.0814027
\(760\) 0 0
\(761\) −11.1005 −0.402393 −0.201196 0.979551i \(-0.564483\pi\)
−0.201196 + 0.979551i \(0.564483\pi\)
\(762\) 0 0
\(763\) −18.3848 −0.665574
\(764\) 0 0
\(765\) −4.41421 −0.159596
\(766\) 0 0
\(767\) −65.3553 −2.35984
\(768\) 0 0
\(769\) −16.5858 −0.598099 −0.299049 0.954238i \(-0.596670\pi\)
−0.299049 + 0.954238i \(0.596670\pi\)
\(770\) 0 0
\(771\) 12.7279 0.458385
\(772\) 0 0
\(773\) −37.1127 −1.33485 −0.667425 0.744677i \(-0.732604\pi\)
−0.667425 + 0.744677i \(0.732604\pi\)
\(774\) 0 0
\(775\) 10.6569 0.382806
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 0 0
\(779\) −11.2132 −0.401755
\(780\) 0 0
\(781\) 4.64466 0.166199
\(782\) 0 0
\(783\) −1.24264 −0.0444084
\(784\) 0 0
\(785\) 1.48528 0.0530120
\(786\) 0 0
\(787\) −45.7696 −1.63151 −0.815754 0.578399i \(-0.803678\pi\)
−0.815754 + 0.578399i \(0.803678\pi\)
\(788\) 0 0
\(789\) 29.7279 1.05834
\(790\) 0 0
\(791\) −0.757359 −0.0269286
\(792\) 0 0
\(793\) −5.79899 −0.205928
\(794\) 0 0
\(795\) 6.41421 0.227489
\(796\) 0 0
\(797\) −27.5269 −0.975053 −0.487527 0.873108i \(-0.662101\pi\)
−0.487527 + 0.873108i \(0.662101\pi\)
\(798\) 0 0
\(799\) 23.8995 0.845503
\(800\) 0 0
\(801\) −10.8284 −0.382604
\(802\) 0 0
\(803\) −34.8873 −1.23115
\(804\) 0 0
\(805\) 1.00000 0.0352454
\(806\) 0 0
\(807\) 18.4142 0.648211
\(808\) 0 0
\(809\) 5.58579 0.196386 0.0981929 0.995167i \(-0.468694\pi\)
0.0981929 + 0.995167i \(0.468694\pi\)
\(810\) 0 0
\(811\) 24.1716 0.848779 0.424389 0.905480i \(-0.360489\pi\)
0.424389 + 0.905480i \(0.360489\pi\)
\(812\) 0 0
\(813\) 23.9706 0.840684
\(814\) 0 0
\(815\) −7.65685 −0.268208
\(816\) 0 0
\(817\) −14.1421 −0.494771
\(818\) 0 0
\(819\) −5.41421 −0.189188
\(820\) 0 0
\(821\) 39.1127 1.36504 0.682521 0.730866i \(-0.260883\pi\)
0.682521 + 0.730866i \(0.260883\pi\)
\(822\) 0 0
\(823\) 38.6274 1.34647 0.673234 0.739430i \(-0.264905\pi\)
0.673234 + 0.739430i \(0.264905\pi\)
\(824\) 0 0
\(825\) 2.24264 0.0780787
\(826\) 0 0
\(827\) 31.0416 1.07942 0.539712 0.841850i \(-0.318533\pi\)
0.539712 + 0.841850i \(0.318533\pi\)
\(828\) 0 0
\(829\) −1.20101 −0.0417128 −0.0208564 0.999782i \(-0.506639\pi\)
−0.0208564 + 0.999782i \(0.506639\pi\)
\(830\) 0 0
\(831\) 11.6569 0.404372
\(832\) 0 0
\(833\) 26.4853 0.917661
\(834\) 0 0
\(835\) 11.0711 0.383130
\(836\) 0 0
\(837\) −10.6569 −0.368355
\(838\) 0 0
\(839\) 40.3431 1.39280 0.696400 0.717654i \(-0.254784\pi\)
0.696400 + 0.717654i \(0.254784\pi\)
\(840\) 0 0
\(841\) −27.4558 −0.946753
\(842\) 0 0
\(843\) −9.07107 −0.312424
\(844\) 0 0
\(845\) 16.3137 0.561209
\(846\) 0 0
\(847\) −5.97056 −0.205151
\(848\) 0 0
\(849\) −24.4558 −0.839322
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) −6.48528 −0.222052 −0.111026 0.993818i \(-0.535414\pi\)
−0.111026 + 0.993818i \(0.535414\pi\)
\(854\) 0 0
\(855\) 7.07107 0.241825
\(856\) 0 0
\(857\) −20.3431 −0.694909 −0.347454 0.937697i \(-0.612954\pi\)
−0.347454 + 0.937697i \(0.612954\pi\)
\(858\) 0 0
\(859\) −34.4558 −1.17562 −0.587809 0.809000i \(-0.700009\pi\)
−0.587809 + 0.809000i \(0.700009\pi\)
\(860\) 0 0
\(861\) 1.58579 0.0540435
\(862\) 0 0
\(863\) 7.85786 0.267485 0.133742 0.991016i \(-0.457301\pi\)
0.133742 + 0.991016i \(0.457301\pi\)
\(864\) 0 0
\(865\) −11.3137 −0.384678
\(866\) 0 0
\(867\) −2.48528 −0.0844046
\(868\) 0 0
\(869\) 12.6863 0.430353
\(870\) 0 0
\(871\) 57.6985 1.95504
\(872\) 0 0
\(873\) −1.17157 −0.0396517
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 12.8284 0.433185 0.216593 0.976262i \(-0.430506\pi\)
0.216593 + 0.976262i \(0.430506\pi\)
\(878\) 0 0
\(879\) 33.0416 1.11447
\(880\) 0 0
\(881\) 25.4142 0.856227 0.428113 0.903725i \(-0.359178\pi\)
0.428113 + 0.903725i \(0.359178\pi\)
\(882\) 0 0
\(883\) −16.3848 −0.551392 −0.275696 0.961245i \(-0.588908\pi\)
−0.275696 + 0.961245i \(0.588908\pi\)
\(884\) 0 0
\(885\) −12.0711 −0.405765
\(886\) 0 0
\(887\) −2.48528 −0.0834476 −0.0417238 0.999129i \(-0.513285\pi\)
−0.0417238 + 0.999129i \(0.513285\pi\)
\(888\) 0 0
\(889\) −11.4142 −0.382820
\(890\) 0 0
\(891\) −2.24264 −0.0751313
\(892\) 0 0
\(893\) −38.2843 −1.28113
\(894\) 0 0
\(895\) 12.3431 0.412586
\(896\) 0 0
\(897\) 5.41421 0.180775
\(898\) 0 0
\(899\) 13.2426 0.441667
\(900\) 0 0
\(901\) 28.3137 0.943266
\(902\) 0 0
\(903\) 2.00000 0.0665558
\(904\) 0 0
\(905\) −6.14214 −0.204171
\(906\) 0 0
\(907\) −1.97056 −0.0654315 −0.0327157 0.999465i \(-0.510416\pi\)
−0.0327157 + 0.999465i \(0.510416\pi\)
\(908\) 0 0
\(909\) 2.75736 0.0914558
\(910\) 0 0
\(911\) −4.97056 −0.164682 −0.0823410 0.996604i \(-0.526240\pi\)
−0.0823410 + 0.996604i \(0.526240\pi\)
\(912\) 0 0
\(913\) 36.0416 1.19280
\(914\) 0 0
\(915\) −1.07107 −0.0354084
\(916\) 0 0
\(917\) −5.65685 −0.186806
\(918\) 0 0
\(919\) −29.9411 −0.987667 −0.493833 0.869557i \(-0.664405\pi\)
−0.493833 + 0.869557i \(0.664405\pi\)
\(920\) 0 0
\(921\) −22.3848 −0.737603
\(922\) 0 0
\(923\) 11.2132 0.369087
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) 0 0
\(927\) 10.4853 0.344382
\(928\) 0 0
\(929\) −20.6152 −0.676364 −0.338182 0.941081i \(-0.609812\pi\)
−0.338182 + 0.941081i \(0.609812\pi\)
\(930\) 0 0
\(931\) −42.4264 −1.39047
\(932\) 0 0
\(933\) −20.9706 −0.686545
\(934\) 0 0
\(935\) 9.89949 0.323748
\(936\) 0 0
\(937\) 45.1716 1.47569 0.737845 0.674970i \(-0.235843\pi\)
0.737845 + 0.674970i \(0.235843\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) 35.5563 1.15910 0.579552 0.814935i \(-0.303228\pi\)
0.579552 + 0.814935i \(0.303228\pi\)
\(942\) 0 0
\(943\) −1.58579 −0.0516403
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) 1.45584 0.0473086 0.0236543 0.999720i \(-0.492470\pi\)
0.0236543 + 0.999720i \(0.492470\pi\)
\(948\) 0 0
\(949\) −84.2254 −2.73407
\(950\) 0 0
\(951\) −3.55635 −0.115323
\(952\) 0 0
\(953\) −57.3137 −1.85657 −0.928287 0.371866i \(-0.878718\pi\)
−0.928287 + 0.371866i \(0.878718\pi\)
\(954\) 0 0
\(955\) 13.8995 0.449777
\(956\) 0 0
\(957\) 2.78680 0.0900843
\(958\) 0 0
\(959\) −0.485281 −0.0156706
\(960\) 0 0
\(961\) 82.5685 2.66350
\(962\) 0 0
\(963\) −3.24264 −0.104493
\(964\) 0 0
\(965\) 7.51472 0.241907
\(966\) 0 0
\(967\) 34.5269 1.11031 0.555155 0.831747i \(-0.312659\pi\)
0.555155 + 0.831747i \(0.312659\pi\)
\(968\) 0 0
\(969\) 31.2132 1.00271
\(970\) 0 0
\(971\) −53.4558 −1.71548 −0.857740 0.514084i \(-0.828132\pi\)
−0.857740 + 0.514084i \(0.828132\pi\)
\(972\) 0 0
\(973\) −2.51472 −0.0806182
\(974\) 0 0
\(975\) 5.41421 0.173394
\(976\) 0 0
\(977\) 40.6985 1.30206 0.651030 0.759052i \(-0.274337\pi\)
0.651030 + 0.759052i \(0.274337\pi\)
\(978\) 0 0
\(979\) 24.2843 0.776129
\(980\) 0 0
\(981\) −18.3848 −0.586981
\(982\) 0 0
\(983\) −0.757359 −0.0241560 −0.0120780 0.999927i \(-0.503845\pi\)
−0.0120780 + 0.999927i \(0.503845\pi\)
\(984\) 0 0
\(985\) 13.7990 0.439672
\(986\) 0 0
\(987\) 5.41421 0.172336
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 24.1716 0.767835 0.383918 0.923367i \(-0.374575\pi\)
0.383918 + 0.923367i \(0.374575\pi\)
\(992\) 0 0
\(993\) 30.7990 0.977376
\(994\) 0 0
\(995\) −23.4558 −0.743600
\(996\) 0 0
\(997\) −40.0833 −1.26945 −0.634725 0.772738i \(-0.718887\pi\)
−0.634725 + 0.772738i \(0.718887\pi\)
\(998\) 0 0
\(999\) 3.00000 0.0949158
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5520.2.a.bk.1.1 2
4.3 odd 2 2760.2.a.q.1.2 2
12.11 even 2 8280.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.q.1.2 2 4.3 odd 2
5520.2.a.bk.1.1 2 1.1 even 1 trivial
8280.2.a.y.1.1 2 12.11 even 2